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Applications of Differentiation (Edexcel IGCSE Maths)
Revision Note
Finding Stationary Points & Turning Points
What is a turning point?
- The easiest way to think of a turning point is that it is a point at which a curve changes from moving upwards to moving downwards, or vice versa
- Turning points are also called stationary points
- stationary means the gradient is zero (flat) at these points
- At a turning point the gradient of the curve is zero.
- If a tangent is drawn at a turning point it will be a horizontal line
- Horizontal lines have a gradient of zero
- This means substituting the x-coordinate of a turning point into the gradient function (aka derived function or derivative) will give an output of zero
How do I find the coordinates of a turning point?
- STEP 1: Solve the equation of the gradient function (derivative / derived function) equal to zero
ie. solve
This will find the x-coordinate of the turning point - STEP 2: To find the y-coordinate of the turning point, substitute the x-coordinate into the equation of the graph, y = ...
- not into the gradient function
Examiner Tip
- Remember to read the questions carefully (sometimes only the x-coordinate of a turning point is required)
Worked example
Classifying Stationary Points
What are the different types of stationary points?
- You can see from the shape of a curve the different types of stationary points
- You need to know two different types of stationary points (turning points):
- Maximum points (this is where the graph reaches a “peak”)
- Minimum points (this is where the graph reaches a “trough”)
- These are sometimes called local maximum/minimum points as other parts of the graph may still reach higher/lower values
How do I use graphs to classify which is a maximum point and which is a minimum point?
- You can see and justify which is a maximum point and which is a minimum point from the shape of a curve...
- ... either from a sketch given in the question
- ... or a sketch drawn by yourself
(You may even be asked to do this as part of a question)
- ... or from the equation of the curve
- For parabolas (quadratics) it should be obvious ...
- ... a positive parabola (positive x2 term) has a minimum point
- ... a negative parabola (negative x2 term) has a maximum point
- Cubic graphs are also easily recognisable ...
- ... a positive cubic has a maximum point on the left, minimum on the right
- ... a negative cubic has a minimum on the left, maximum on the right
Worked example
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